A Robbins–Monro procedure for estimation in semiparametric regression models

Abstract

This paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model. We implement a very efficient and easy to handle Robbins-Monro procedure. On the one hand, we propose a stochastic algorithm similar to that of Robbins-Monro in order to estimate the shift parameter. A preliminary evaluation of the regression function is not necessary to estimate the shift parameter. On the other hand, we make use of a recursive Nadaraya-Watson estimator for the estimation of the regression function. This kernel estimator takes into account the previous estimation of the shift parameter. We establish the almost sure convergence for both Robbins-Monro and Nadaraya-Watson estimators. The asymptotic normality of our estimates is also provided. Finally, we illustrate our semiparametric estimation procedure on simulated and real data.

Recursive estimation in a class of models of deformation

“…Firstly, (6.28) of [1] together with the almost sure convergence of a n to a as n goes to infinity, lead to…”

“…For all 1 ≤ i ≤ n, the noise (ε i,j ) is a sequence of independent random variables with mean zero and variances E ε 2 i,j = σ 2 j , and independent of the random points X i . In addition, as in [1], we make the following hypothesis.…”

“…Roughly speaking, we are going to implement our Robbins-Monro procedure as in [1] for each component of θ.…”

“…The case where the X i are equi-distributed deterministic variables has been considered in [7], [21] or [23]. When p = 1, a 1 = 1 and v 1 = 0, Bercu and Fraysse [1] propose a recursive method to estimate the translation parameter θ 1 . In this paper, we significantly extend their results as we are now able to estimate, whatever the value of the dimension parameter p is, the height parameter v, the translation parameter θ and the scale parameter a, respectively given by…”

A parametric registration model for warped distributions with Wasserstein’s distance

“…These deformations prevent the use of usual methods in data analysis and deserve a special statistical treatment in order to align the data. We refer for instance to (Gamboa et al, 2007), (Dupuy et al, 2011), (Ramsay and Silverman, 2005), (Bercu and Fraysse, 2012) and references therein to applications to functional data analysis, to (Trouve and Younes, 2005) or (Amit et al, 1991) for applications in image analysis or to (Bolstad et al, 2003) for applications in biology.…”